Introduction to Particle-in-Cell Method in Plasma Simulations

Transcription

1 Introduction to Particle-in-Cell Method in Plasma Simulations Chuang Ren University of Rochester The 2009 HEDP Summer School Acknowledgement: V. Decyk, R. Fonseca, W. Mori, J. Tonge, F. Tsung 1

2 Simulation Is an Important Tool in Scientific Research Computer simulations are carried out to Understand the consequences of fundamental physical laws Experiment Help interpret experiments Can provide detailed information which is difficult to measure Design and predict new experiments Simulation Theory 2

3 PIC simulations are indispensible in HEDP research Outline What is PIC method? Numerical algorithm for 2-1/2D EM code An example: fast ignition 3

4 There Are Various Ways to Describe a Plasma The complete description is to specify x(t) & v(t) of each particle Klimontovich formalism Neglecting particle-correlation (collisions), we get Vlasov description Closed Vlasov-Maxwell eqs. Fluid description describes macroscopic quantities Closure problem (For details see textbooks, e.g. Krall & Trivelpiece) [ t + v + q α (E + v B ) v ] f α ( x, v,t) = 0 m α c E = 4π q α f α dv B = 1 c E = 1 c α E t + 4π c B t α q α n α ( x,t) + (n α V α ( x,t)) = 0 t n α m α v f α dv t V α + n α m α V α V α n α q α (E + V B α c = n α m α (V α V β ) < ν αβ > β ) + P 4

5 Particle Simulations Takes Klimontovich Approach Particle methods simulate a plasma system by following a number of particle trajectories Essential physics can be captured with a much smaller number of particles than that in a real plasma Charge/mass ratio and charge density remains the same 10 3 ~10 11 particles used Statistics of the model system is different from a real plasma The detailed description is computation-intensive Maxwell s Eqs x i, v E( i x i ),B( x i ) Newton s Eq + Lorentz force ω p 2 = 4πne2 m Invariant e/m and ne -> invariant ω p 5

6 Phase-space Is Efficiently Represented in Particle Methods It is difficult to sample 6Dphase space by cells o Difficult to solve Vlasov equations in high dimensions P unoccupied phase space (wasted cells) o PIC codes use particles to efficiently sample the phase space distribution plasma distribution function X 6

7 Calculating Inter-particle Force Directly Is Expensive number of calculations~ 10 N p particles, 10 Tflop/s 1 time step ~ 3 hrs time step sim. ~ 3 yrs 7

8 Particle-in-Cell Algorithms Find (ρ ij,j ij ) from (x k,v k ) Solving Maxwell s eqs on grid to obtain (E ij,b ij ) Find F k from (E ij,b ij ) Number of calculatons for 1 step Deposit charge/current N p Advance fields N cells Interpolate forces and advance particles N p Total ~ αn p + β(n cells ) 10 8 particles, 10 Tflop/s 64 x 64 x 64 cells 1 time step ~ 0.3 ms time step sim. ~ 3 s 8

9 The Simple & Straightforward PIC Scheme dp dt = q E + v c B Integration of equations of motion, moving particles F k u k x k Weighting ( E, B ) ij F k Δt Weighting (x,u) k J ij Integration of Field Equations on the grid ( E, B ) ij J ij E = 4π j c B t B t = c E 9

10 Finite-size Particles Reduce Collision Close-range collisions are much reduced by using the finite-size particles This compensates for the smaller number of particles used ν/ω p ~(nλ D3 ) -1 We are interested in nλ D3 >>1 regime PIC models are not collisionless Can be used to test collision operators 10

11 References on PIC methods J. M. Dawson, Particle simulation of plasmas, Rev. Mod. Phys. 55, (1983) R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York (1981) C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Institute of Physics Publishing, Bristol, UK (1991) 11

12 Numerical Algorithms for a 2-1/2D Electromagnetic PIC Code no spatial variation in z (half-dimension) 12

13 Advancing EM Fields The EM fields are advanced by Maxwell s Equations The other 2 Maxwell s eqs are satisfied as initial conditions In 2D, the field components can be decomposed into 2 independent modes TE mode (k E=0): E z, B x, B y. t B = E, t E = B J. E = ρ, B = 0. TM mode (k B=0): B z, E x, E y. t B x = y E z, t B y = x E z, t E z = x B y y B x J z. t B z = ( x E y y E x ), t E x = y B z J x, t E y = x B z J y. 13

14 Centered Difference Can Be Used in Both Time & Space Domain For TM mode n +1/ B 2 n 1/ 2 zi, j B zi, j n +1 E xi, j+1/2 n +1 E yi+1/2, j Δt n E xi, j+1/2 Δt n E xi+1/2, j Δt = E n n y i+1/2, j E yi 1/2, j Δx = B n +1/ 2 z i, j+1 B zi, j Δy n +1/ 2 = B n +1/ 2 z i+1, j B zi, j Δx TE mode is similar n +1/ 2 + E n n x i, j+1/2 E xi, j 1/2 Δy j xi, j+1/2 j yi+1/2, j j (E x,, J x ) B z (E y,, J y ) i B x j B y (E z, J z ) i 14

15 Combining Two Sets of Fields Two equal ways of putting both modes on one grid Yee lattice j+1 j+1/2 j (E z,, B z, J z ) (E y,, B y, J y ) i (E x,, B x, J x ) or i+1/2 i+1 Leapfrog in time domain j+1 j+1/2 j (E z,, J z ) (E x,, B y, J x ) (E y,, B x, J y ) B z i i+1/2 i+1 E B,J n-1 n-1/2 n n+1/2 x V 15

16 Time Step Is Limited By Courant Condition Plane EM Waves in Vacuum (E,B)=(E 0,B 0 )exp(ik x-iωt) Dispersion relation from difference scheme Small t is required for stability in explicit schemes Large t can be achieved with implicit schemes Still need to resolve relevant physics ΩB = κ E ΩE = κ B Ω 2 = c 2 κ 2 sin ωδt 2 cδt 2 = sin(ωδt /2) Ω = ω ωδt /2 sin(k κ x = k x Δx /2) x k x Δx /2 sin k x Δx 2 Δx c 2 Δt > 1 2 Δx Δy 2 sin k y Δy 2 Δy Courant Condition for Numerical Stability 2 16

17 Numerical Dispersion Relation Allows Modes with V ph <C ωδx/c ω=kc V ph =0.68c For relativistic particles with V>V ph, unwanted Cerenkov emission can occur. 2 1 Numerical dispersion (c t/ x=0.6) kδx sin ωδt 2 cδt 2 = sin k x Δx 2 Δx 2 + sin k y Δy 2 Δy 2 17

18 Alternative Field Solver Using FFT: Spectral Codes Transform all quantities to k- space Decompose (j,e) into transverse & longitudinal components Updating fields in k-space FFT (E,B) back to find forces on particles Vacuum dispersion relation V ph >c for small k, no spurious Cerenkov radiation High k-modes can be truncated fft ρ(x), j(x), E(x),B(x) ρ(k), j(k),e(k),b(k) j L ( k k k j ( k ) ) =, j T ( k ) = j ( k ) j L ( k ) k 2 B T (k,t) = ik E T (k,t) t E T (k,t) = ik B T (k,t) j T (k,t) t ike L (k,t) = ρ(k,t) k 2 c 2 = 4 ωδt Δt 2 sin2 2 k 2 c 2 Δt 2 <1 (Courant cond.) 4 ω = ±kc(1+ k 2 c 2 Δt ) 24 18

19 Pushing Particles To update p, EM fields have to first be interpolated from the grid to the particle locations Linear (area) weighting (i, j) ΔA Δx Δy E( x k ) = E ij S( x k X ij ) i, j S( x k X ij ) = ΔA ΔxΔy S( x k X ij ) =1 i, j Same S should be used in charge deposition to ensure momentum conservation dp/dt=0 for 1 particle, independent of S. ρ ij = q k S( x k X ij ) k dp dt = q E( k x k ) = q k E ij S( x k X ij ) k k i, j = E ij i, j ρ ij 1 2 E 1 = ρ 2 2,E 2 = ρ 1 2 E 1ρ 1 + E 2 ρ 2 = 0 19

20 Pushing Particles (Boris Pusher) Differencing Relativistic Newton s Eq. p n +1/ 2 p n 1/ 2 Δt = q m (E n + 1 p n +1/ 2 + p n 1/ 2 B n ) c 2γ n (B n = Bn +1/ 2 + B n 1/ 2 ) 2 Separating f E & f B The magnetic force causes a pure rotation p = p n 1/ 2 + Δt 2 p n +1/ 2 = p + + Δt 2 p + p Δt = q mc [(p + ) 2 ( p ) 2 = 0] q m E n q m E n p + + p _ B n 2γ n 20

21 v B Rotation Eq. for v B rotation p + p Δt = q mc p + + p _ 2γ n B n p + θ p + p p + p + tan θ 2 = qb 2mcγ Δt p In practice, this is used p'= p + p t, (t = qbδt /2γmc = tan θ 2 ) p + = p + p' s, (s = 2t /(1+ t 2 ) = 2sin θ 2 cosθ 2 ) p + θ p' p' s p p t 21

22 Current Deposition & Charge Conservation Updating positions Current deposition needs both V & x defined at different times ρ/ t+ j=0 may not be satisfied Charge conservation is critical to guarantee solutions satisfying all Maxwell s eqs. Two ways to achieve charge conservation Adjusting longitudinal E Using charge-conserving current deposition schemes Spectral method x n +1 = x n + pn +1/ 2 Δt n +1/ 2 γ x n +1 = x n + j ij n +1/ 2 = q k v k n +1/ 2 S( or k j ij n +1/ 2 = q k v k n +1/ 2 k pn +1/ 2 Δt n +1/ 2 γ x n k + x n +1 k t B = E t B = 0 2 X ij ) S( x k n X ij ) + S( x k n +1 X ij ) t E = B J t E = J 2? t ρ 22

23 Adjusting E to Satisfy E=ρ E obtained from a non-cc j does not satisfy E =ρ Find δφ so that 2 δφ= E -ρ The corrected field is E=E - δφ j+1 j+1/2 j (E y,, B y, J y ) (ρ,δφ) i (E z,, B z, J z ) (E x,, B x, J x ) i+1/2 i+1 23

24 Charge-Conserving Current Deposit Scheme For unit grid & unit charge J x1 = Δx (0.5 y) + (0.5 y Δy) 2 J x2 = Δx(0.5 + y Δy) J y1 = Δy(0.5 x 1 2 Δx) J y2 = Δy(0.5 + x Δy) = Δx(0.5 y 1 2 Δy) For more complicated crossing, the current can be calculated in stages [see Villasenor & Buneman, Computer Phys. Comm. 69, 306 (1992).] 24

25 Boundary Conditions For fields Periodic Conducting Open, absorbing, For particles Periodic Reflecting Thermal bath z E 1/ 2, j y E 0, j z z = E Nx 1/ 2, j,e Nx+1/ 2, j = 0, E z 1/ 2, j z + E 1/ 2, j 2 = 0 z = E 1/ 2, j,... j+1 j+1/2 j (E z,, B z, J z ) (E y,, B y, J y ) i (E x,, B x, J x ) i+1/2 i+1 25

26 Grid Instability Since particle positions are continuous, δn can have high-k modes Dominated by k n ~ω p /v t Due to aliasing, δρ can have modes with k=k n -p(2π/ x) δρ can resonantly interact with E of the same k, causing instability In practice, if <3λ D, the grid instability is largely suppressed Smoothing also helps 26

27 Parallelization: Domain Decomposition Simulation Volume 1D 2D 3D Divide simulation space between multiple cpu s 1 x cpu 4 x cpu 16 x cpu 64 x cpu Use guard cells to store needed information from neighboring nodes node 1 node 2 node 1 node 2 guard cells Node Boundaries Communication 27

28 Parallelization II Distributed memory systems using MPI Field solver is local: Communication only with neighboring nodes For parallelizing the global FFT, see Decyk, Comp. Phys. Comm., 1994 Same decomposition for particles and fields Each computing node acts as a section of the global physical space All parallelization is encapsulated in boundary condition routines Developers can focus on local physics Base classes for grid quantities include boundary-to-another-node routines 28

29 osiris loop (~1µs/particle-step on G5 xserve) Sample 2 node run. The arrows indicate communication between the two nodes diagnostics update emf bound advance& deposit update particle bound update j bound current smooth field solver 0 Node t [s]

30 Other kinds of PIC codes Hybrid codes Fluid description for background plasma + kinetic description for energetic particles Implicit field solver Allows larger Δt Not resolving physics <Δt More difficult to scale up to large # of processors 30

31 Summary on PIC codes The PIC model is a first-principle kinetic model for plasma physics It efficiently tracks particle trajectories to sample particle phase space Its structure is scalable to massively parallel computer systems 31

32 PIC Simulations for the Channeling/Hole-Boring Scheme in Fast Ignition 32

33 Separating Compression and Heating: Fast Ignition ρ=100 s g/cm 3 achievable from compression Compressed fuel Difficult to form hot-spot in conventional ICF Using a second ignition laser relaxes compressing laser requirement Analogy: diesel engine vs gasoline engine Heating window: 10 ps ignition laser e - Key issues: Ignition pulse propagation Hot e - generation e - transport e - stopping in dense matter overdense plasma

34 PIC Is Suitable to Study a Large Fraction of FI Targets FI targets have a large lowcollisionality region Kinetic description important PIC is preferred to study Laser-plasma interactions Hot e - generation & transportation The physics can only be studied in scaled-down simulations data courtesy of R. Betti, plot by J. Tonge

35 FI simulations requires tremendous computation resources For explicit PIC 3D simulations, Total memory scales as L 3 n 3/2 Total particle-step scales as L 3 Tn 2 Ex. 1: laser channeling in 1 mm underdense plasma 1000µm (200µm) 2 box for 100ps requires grids and 10 6 time steps 6.4 trillion particles, 360 TB memory, and a total of FLOPs (9 days on a peta-flop machine) Ex. 2: e- generation and transport in 100 n c plasma (500µm) 3 box for 2 ps requires grids and time steps particles, 170 PB memory, and a total of FLOPs (20 days on a exa-flop machine) 35

36 Ex.1: Channeling reduces energy loss for the ignition pulse in fast ignition High-intensity ignition pulse can lose energy in the mm underdense corona of the FI targets P/P c ~10 6 P(PW) n/n c >>1 channeling/hole-boring pulse Two ways of avoiding loss in corona Cone targets ignition pulse Key questions Channeling/hole-boring Symmetric implosion Avoid issues associated with a gold cone Requires an ultra-intense ignition pulse Can laser create a straight channel? What is the channeling speed? What is the optimum intensity for the channeling pulse Density- and intensity-scalings

37 Laser channeling in mm scale plasmas is a highly nonlinear and dynamic process Previous experiments and simulations on channeling used 100-µm plasmas Full-scale 2D simulations with the PIC code OSIRIS 1000µm 400µm plasmas Plasma density rises from 0 to 1 n c in ~ 1000 µm exponentially M i /m e =4590 (DT), T i =T e = 1 kev λ=1 µm, I= W/cm 2, W 0 =14-40 µm, t>10 ps Reduced-scale 3D simulations [up to 540µm (90µm) 2 plasma, 17 billion particles] Ref: Li et al, PRL, 100, (2008)

38 Stages of channeling process Relativistic SF/Filament little density modification a increases and w decreases in each filaments n i E laser n e

39 Stages of channeling process Relativistic SF/Filament Ponderomotive SF/Filament Micro channels created from laser filaments n i E laser n e

40 Stages of channeling process Relativistic SF/Filament Ponderomotive SF/Filament Central filament widening & shock launching Laser snowplows away micron channel walls to form a single channel Transverse expansion through high-mach-number shocks V t ~0.03c ~2C s (at 500 kev) Channel wider than laser width Laser hosing/channel branching seen E laser n i

41 Channel advancing is intermittent Longitudinal advancing involves many highly nonlinear processes Plasma snowplow Density buildup can reach n c

42 Channel advancing is intermittent Longitudinal advancing involves many highly nonlinear processes Plasma snowplow Laser hosing/channel bending Channel bifurcation Laser breaks out from channel wall Channel self-correction Laser blows away density island Li et al., PRL 2008

43 Relativistic Te reduces nonlinear interactions PIC simulations show the residual electrons are quickly heated to relativistic Te When Te is relativistic, the quiver velocity v z is reduced*: v os = a/γ (1 + 5p th 2 ) 1/ 2 The ponderomotive force is reduced, leading to the laser-plasma decoupling y(µm) Time t=0.56ps x(µm) Te(MeV ) Decoupling at relativistic temperature increases the laser transmission *K. C. Tzeng et al., Phys. Rev. Lett. 81, 104 (1998).

44 3D simulations have also shown the same nonlinear and dynamic phenomena 3D simulations [up to 540µm (90µm) 2 plasma, 17 billion particles] 0.0 z (µm) Charge density (ec -2 ω p 2 ) Laser hosing/channel bending & branching/self-correction seen in 3D but at slower growth y (µm) -1.0 The eventual channel cross section is round Observed in RAL experiments (Norreys, 2009 Omega User Workshop)

45 3D simulations show a larger channeling speed than in 2D 3D Phasespace Time t=3.0ps 2D The average residual density in 3d is smaller v c-3d =0.24c v c-2d =0.13c V 3D =2 V 2D due to stronger laser selffocusing and easier channeling in 3D Channel front position(µm) 3D 2D Time(ps)

46 The stonger 3D ponderomotive force allows the channel to be formed faster n i /n cr 0.35 For the same laser ponderomotive force F p ~a 2 /w, the channel is deeper in 3D than in 2D Fp is larger in 3D than in 2D due to self-focusing a 2 W 2 = const 3D: 2D: Fp3d/Fp2d = w0/wf ~ 2 vt=1kev ni 0 =0.3n cr w 0 =90 a 0 =2.7 t=0.84ps a 2 W = const D 3D y(µm)

47 Lower intensity pulse reduces channeling energy 2D Simulation scaling T c =290I ps & E c =1.7I kj 3D results indicate T c & E c may be halved For I 18 =2, T c =93 ps & E c =1.1kJ In the OMEGA/EP parameter range We need to understand laser hosing in the relativistic, long-pulse regime

48 Laser channeling in mm-scale plasmas is a highly nonlinear and dynamic process A low-density channel can significantly increase the transmission of the ignition pulse in FI Residual electrons are heated to relativistic temperatures, which reduces laser-plasma coupling in the channel Lower-intensity pulses reduce total energy for channeling LPI of kj, sub-ns lasers are in a new regime where ion motion and electron relativistic effects are both important

49 Ex.2: e- generation and transport depend on global e - trajectories Significant e - refilling is needed to absorb laser energy ηit = (nδe) L L~40 µm for I=10 20 w/cm 2, t=1ps, and n=40n c Hot e - distribution will be affected by simulation size and boundary conditions Ref.: Ren et al., PRL, 93, (2004); Tonge et al., Physics of Plasmas 16, (2009) L 49

50 Setup of OSIRIS simulations Vacuum region between target and boundary grids (Δx=0.33 c/ω p ), with current smoothing particles and steps Initial Te=7.4 kev and Ti=1 kev 1µm-laser, I= w/cm 2, spot size (FWHM) 7.5 µm, 1 ps long, s- & p- polarized. laser coronal plasma 40n c 16µm 100µm 25.5µm 50

51 A movie of e - density (overview) 51

52 Laser-plasma Interface shows complex & dynamic features 52

53 Laser absorption efficiency changes dynamically Absorption increases as critical surface ripples P 2 E 2 vs resonant & vxb heating Absorption rate does not change significantly as intensity increases 53

54 Hot e - Have Significant Transverse Spread p2p1 phase space of e - in front of laser (t=454fs) <γ-1>=1.4, ΔP 1 /mc=2.9, ΔP 2 /mc=2.0 54

55 Distribution functions The beam is by no means a bump on the tail of a gaussian. The distribution function of the electrons moving forward may be approximated by a cold Maxwellian and a power law. By fitting the tail we obtain: #particles de ~ 1/E (2-3) ~3decades Initial Maxwellian ( all particles that move forward) 55

56 e - Transport Is Influenced by Magnetic Fields >1 MeV e - distribution B3-field near L-P interface 56

57 Applying the theory to simulations 681fsec Electron denstiy M e a s u r e d d i s t r i b u t i o n function is divided into many beams Ion mode Marginal instability found in the shock region Return current contribution important Outside the shock the mode is found stable p Stable 57

58 No Global Filaments Merging Is Seen This is consistent with the e - distribution which is stable to Weibel outside shock region Magnetic force can t overcome thermal pressure 58